Uniform approximation of fractional derivatives and integrals with application to fractional differential equations

Abstract

It is well known that for every $f\in C^m$ there exists a polynomial $p_n$ such that $p^{(k)}_n\rightarrow f^{(k)}$,
$k=0,\ldots,m$. Here we prove such a result for fractional
(non-integer) derivatives. Moreover, a numerical method is proposed for fractional differential equations.  The convergence rate and stability of the proposed method  are obtained. Illustrative examples are discussed.